Colloquia

Summary

The Saffman lift force, which describes the lift experienced by a sphere in a viscous shear flow, is widely cited and applied across various fields, including sediment transport, pharmaceuticals, blood flow, bubble physics, and microfluidics. The lift force is generally accepted as given by Saffman. However, Saffman’s original derivation (1965) is mathematically intricate, concise to the point of opacity, and inconsistent in notation, making it difficult to follow, extend, and develop further. Notably, no published mathematical review of the derivation has been identified. This thesis revisits Saffman’s analysis by providing a detailed, step-by-step derivation of the zeroth-order and first-order perturbation expansions of the governing fluid mechanics equations. At zeroth order, three distinct solutions for velocity, pressure, and force emerge, differing only by a constant factor and thus exhibit similar behavior. The three distinct solutions arise from Saffman's original derivation, Lamb's earlier formulation (1932), and a novel derivation presented in this thesis, incorporating an alternative approach based on Lamb’s earlier formulation, which, upon closer examination, resulted in a subtle discrepancy. At first order, the derived force expression depends on the zeroth-order velocity but introduces two unknown vectors. Additionally, a framework is established for extending the theory to related problems, such as the lift force on a particle in a viscous shear flow of granular media.